**4. Realization of proposed approach**

Let *m* experts evaluate the values of *n* parameters in the form of trapezoidal fuzzy numbers. As a result, we get a rectangular matrix of dimension *m* � *n*:

$$\begin{array}{ccccccccc}\tilde{R}\_{11} & \tilde{R}\_{12} & \cdots & \tilde{R}\_{1n} & & & & \\ & \tilde{R}\_{21} & \tilde{R}\_{22} & \cdots & \tilde{R}\_{2n} & & & \\ & \cdots & \cdots & \cdots & \cdots & & & & \\ & & \tilde{R}\_{m1} & \tilde{R}\_{m2} & \cdots & \tilde{R}\_{mn} & & & \end{array} , \ i = \overline{1, n}, j = \overline{1, n}. \tag{17}$$

The *i-*th column of the obtained matrix represents a collection of estimates of the *j*th parameter given by the expert number *i*.

To assess the values of the parameters according to the credit risk criterion, we find a representative of each column of the matrix. As a result, we obtain the following sequence: *R*~ <sup>∗</sup> *j* n o <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>1</sup> , *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>2</sup> , … , *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> *n :*

Now we present an algorithm for finding a representative of finite collection of trapezoidal fuzzy numbers.

#### **Algorithm 1.**

A representative can be regarded as a kind of group consensus, but in that case the degrees of experts' importance are neglected. A representative is something like a standard for the members of the considered collection. As the weights of physical bodies are measured by comparing them with the Paris standard kilogram, it seems natural for us to determine experts' weights of importance depending on how close

Thus, the main idea of the proposed method reduces to the following. The weight of importance for each expert is determined by a function inversely proportional to the distance between his estimate and the representative of the finite collection of all experts' estimates, i.e., the smaller the distance between an expert's

Let *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>*, *<sup>j</sup>*∈f g 1, 2, … , *<sup>m</sup>* , *<sup>m</sup>* <sup>¼</sup> 2, 3, … be a trapezoidal fuzzy number representing

� �. By Definition 4 and formula (14), we find the regulation *R*~<sup>0</sup>

By the above reasoning, the weights and the final result of aggregation can be

, *R*~ *<sup>j</sup>*

*ω<sup>j</sup>* ⊗ *R*~ *<sup>j</sup>*

� � � � �<sup>1</sup> , *<sup>m</sup>* <sup>¼</sup> 2, 3, … (15)

*<sup>j</sup>*¼<sup>1</sup>*ω<sup>j</sup>* <sup>¼</sup> 1. In [7] it is proved that the function in expression

� � (16)

, *R*~ *<sup>j</sup>* � �

.

the representative *R*~ <sup>∗</sup> of this collection. Denote the *j*th expert's aggregation weight (weight of importance) and the final result of aggregation by *<sup>ω</sup><sup>j</sup>* and *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>a</sup>*~, <sup>~</sup>

*j* n o

and

*b*,~*c*, ~ *d* � �

� �,

<sup>¼</sup> 0, *then <sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup>

.

,

estimate and the representative, the larger the weight of his importance.

*ρ R*~ <sup>∗</sup> , *R*~ *<sup>j</sup>* � � � � �<sup>1</sup>

*<sup>j</sup>*¼<sup>1</sup> *<sup>ρ</sup> <sup>R</sup>*<sup>~</sup> <sup>∗</sup>

*<sup>R</sup>*<sup>~</sup> <sup>¼</sup> <sup>X</sup>*<sup>m</sup> j*¼1

(15) is always continuous (the denominator does not turn into 0 in any case).

**Proposition 1** [7]**.** *For any finite collection of trapezoidal fuzzy numbers R*~ *<sup>j</sup>*

In the following proposition and its corollaries, the properties and values of the

� � *is always continuous* (*here ωj is given by Eq.* (15)).

**Summary**. The section introduces the approach to assessment of the credit risk.

It also considers the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts' estimates. The section contains important formalisms for determining the degrees of experts' importance and the result of

the *j*th expert's subjective estimate of the rating to an alternative under a given criterion. Estimates of all experts form the finite collection of trapezoidal fuzzy

experts' estimates are to a representative.

*ω<sup>j</sup>* ¼

aggregation result for special cases are established.

b. *If there exists at least one j* ∈ {1,2, … ,*m*} *such that ρ R*~ <sup>∗</sup>

**Corollary 1.** *If for all t, j* <sup>∈</sup> {1,2, … ,*m*} *<sup>R</sup>*~*<sup>t</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>* ) *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup>

**Corollary 2.** *If the all estimates are identical then ω<sup>j</sup>* = 1/*m*.

*j* ¼ 1, *m*, *m* ¼ 2, 3, … , *the following holds:*

*<sup>j</sup>*¼<sup>1</sup> *<sup>ω</sup><sup>j</sup>* <sup>⊗</sup> *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>*

aggregation of experts' estimates.

P*<sup>m</sup>*

numbers *R*~ *<sup>j</sup>*

*Banking and Finance*

respectively.

and

defined as follows:

It is obvious that P*<sup>m</sup>*

a. *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> <sup>P</sup>*<sup>m</sup>*

**188**

Step 0: *Initialization: the finite collection of trapezoidal fuzzy numbers R*~ *<sup>j</sup>* � �*, its regulation R*~<sup>0</sup> *j* n o, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … . *Denote the aggregation weight of the j-th expert by ω<sup>j</sup> and the final result by R*~.

Step 1: *Compute the representative R*~ <sup>∗</sup> *of R*~<sup>0</sup> *j* n o, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … *by Eq.* (14)*.* Step 2: *Do* Step 3 *for j* ¼ 1, *m.* Step 3: *Compute* <sup>Δ</sup> *<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>R</sup>*<sup>~</sup> <sup>∗</sup> , *R*~ *<sup>j</sup>* � �:


For what follows, we need to determine a scale that can "measure" the opinions of experts regarding the risk of bankruptcy of the borrower. We use the general approach described in [11].

In almost any field, you can get a rating scale using the following principles:


The collection of ratings on the scale was called the *profile* of the concept. Since the gradation values of the scale are approximate (expert opinions), with the exception of the assigned pole values, the profile represents a fuzzy set of the concept's characteristics list.

In the introduction, we mentioned the concept of a linguistic variable. This concept plays an important role in our study. Let us introduce the linguistic variable "degree of credit risk":

$$\mathbf{A} = \{A\_1, A\_2, A\_3, A\_4, A\_5\},\tag{18}$$

It is easy to see that the coefficient of proportionality between the abscissas of

Thereby, the coordinates of the original trapezoidal fuzzy numbers will change

*<sup>A</sup>*~<sup>3</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>t*<sup>2</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>2</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>3</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>t*<sup>4</sup> <sup>þ</sup> <sup>∇</sup> ; *<sup>A</sup>*<sup>~</sup> <sup>4</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>t*<sup>3</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>3</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>4</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>t*<sup>5</sup> <sup>þ</sup> <sup>∇</sup> ;

We continue the description of the implementation of the proposed approach. Based on Algorithm 1, we find the value of the representative of the finite collection of the trapezoidal fuzzy numbers for each parameter. Consider a representative calculated for the *i-*th parameter. The risk assessment threshold value is assigned by the manager (group of managers) of the lender. It may be that different criteria thresholds will be set for different cases, for example, for one parameter, "no more than *A*2—the degree of risk is low," and for the other "no more than *A*3—the degree

In general, if *Aj* ∈ *A* (see Eq. (18)) is taken as the threshold criteria value of the parameter, then credit risk is acceptable if the following addition condition is fulfilled:

Here *A*~ *<sup>j</sup>* is the number corresponding to the characteristic *Aj*, while *R*~*<sup>i</sup>* is the result of aggregation of the finite collection of expert estimates of the *i-*th parameter

Let us summarize the above as a generalized algorithm. So, we have the follow-

result of aggregation *R*~*<sup>i</sup>* of the finite collection of expert estimates for this parameter, the threshold criteria value *Ak*, *k* ¼ 1, 5 selected from Eq. (18), and coordinates *k, t* specified by the manager (group of managers) of the lender for use in Eq. (23).

Step 0: *Initialization: fix pi* <sup>∈</sup> *<sup>P</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, *the result of aggregation <sup>R</sup>*~*<sup>i</sup> of the finite*

ing input: *m* expert estimates of *n* parameters out of the set *P* ¼ *pi*

*threshold criteria value At*, *t* ¼ 1, 5, *coordinates*: *k*1*, … ,k*4; *t*1*, … ,t*5. Step 1: *Compute* ∇, *Δ by Eq.* (21) *and λ by Eq.* (22).

• If the condition is met then the level of risk is acceptable;

• If the condition is not met then the level of risk is unacceptable.

**Summary.** The section looks at the realization of the proposed approach. The linguistic variable "degree of credit risk," polar percentage and coordinate scales are formed. The criterion for an assessment of the credit risks is generated. This section also presents two generalized algorithms for implementing the

*<sup>R</sup>*~*<sup>i</sup>* <sup>⪯</sup> *<sup>A</sup>*<sup>~</sup> *<sup>j</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 5*:* (24)

, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … *for this parameter, the*

*<sup>A</sup>*~<sup>1</sup> <sup>¼</sup> ð Þ <sup>∇</sup>, <sup>∇</sup>, *<sup>λ</sup>k*<sup>1</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>t*<sup>2</sup> <sup>þ</sup> <sup>∇</sup> ; *<sup>A</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>t*<sup>1</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>1</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>2</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>t*<sup>3</sup> <sup>þ</sup> <sup>∇</sup> ;

*λ* ¼ 0*:*01ð Þ *Δ* � ∇ *:* (22)

(23)

, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, the

the primary and the new coordinate system is

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

*A New Approach for Assessing Credit Risks under Uncertainty*

as follows:

*<sup>A</sup>*~<sup>5</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>t*<sup>4</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>λ</sup>k*<sup>4</sup> <sup>þ</sup> <sup>∇</sup>, *<sup>Δ</sup>*, *<sup>Δ</sup> :*

of risk is medium."

(see Algorithm 1).

**Algorithm 2.**

proposed approach.

**191**

*collection of expert estimates R*~ *<sup>j</sup>*

Step 2: *Compute <sup>A</sup>*~*t*, *<sup>t</sup>* <sup>¼</sup> 1, 5 *by Eq*. (23). Step 3: *Verification of the condition R*~*<sup>i</sup>* ⪯ *A*~ *<sup>j</sup>*:

So, condition (c) is also satisfied.

where:

*A*<sup>1</sup> – the degree of risk is negligible.

*A*<sup>2</sup> – the degree of risk is low.

*A*<sup>3</sup> – the degree of risk is medium.

*A*<sup>4</sup> – the degree of risk is high.

*A*<sup>5</sup> – the degree of risk is extreme.

Therefore, by constructing the linguistic variable *A*, we have satisfied condition (a). Condition (b) is also fulfilled: the polar characteristics are "the degree of risk is negligible" and "the degree of risk is extreme." To fulfill condition (c), it is necessary to build a profile, i.e., fuzzy set describing the linguistic variable *A*.

We construct the membership function of profile *A* in several stages. Here we will give a description of the stages in a general form; the reader will clarify the specifics on a practical example in the next section.

**Stage 1.** Let us evaluate the confidence of risk degrees of the linguistic variable *A* on the percentage scale (0–100)% as follows: *A*<sup>1</sup> � [0, *k*1], *A*<sup>2</sup> � [*k*1, *k*2], *A*<sup>3</sup> � [*k*2, *k*3], *A*<sup>4</sup> � [*k*3, *k*4], and *A*<sup>5</sup> � [*k*4,100]. Here 0 < *k*<sup>1</sup> < *k*<sup>2</sup> < *k*<sup>3</sup> < *k*<sup>4</sup> < 100.

**Stage 2**. Since expert estimates are given in the form of trapezoidal fuzzy numbers, first of all, it is necessary to determine the boundaries of the scale of expert estimates for each characteristic of the list from *A*. Since *m* experts take part in the assessment process, we have *m* trapezoidal fuzzy numbers. It seems reasonable to take the following boundaries of the scale: the left one is the minimum, and the right maximum of the abscissas of all the vertices of *m* trapezoids, i.e.:

$$[\min\left\{a\_i\right\}; \max\left\{d\_i\right\}], \ i = 1, m. \tag{19}$$

**Stage 3.** Now we will establish the conformity between the intervals of the percentage scale and the trapezoidal fuzzy numbers. Geometrically, the percentage scale that corresponds to five trapezoidal fuzzy numbers, may, for example, look like this (**Figure 4**):

**Figure 4.** *Conformity between the percentage scale and the trapezoidal fuzzy numbers.*

$$\begin{aligned} \tilde{A}\_1 &= (0, 0, k\_1, t\_2); \tilde{A}\_{22} = (t\_1, k\_1, k\_2, t\_3); \tilde{A}\_3 = (t\_2, k\_2, k\_3, t\_4); \tilde{A}\_4 = (t\_3, k\_3, k\_4, t\_5); \\ \tilde{A}\_5 &= (t\_4, k\_4, 100, 100), \end{aligned} \tag{20}$$

numbers *k, t* are appointed by experts.

To transform the coordinate system of the percentage scale to the coordinate system for expert estimates, the following mappings should be performed ½ �! 0, 100 min f g *ai* , max f g *di* ½ �*:* Thus, we moved the origin from point (0,0) to point ( min f g *ai* , 0) and point (100,0) to point max f g *di* ð Þ , 0 *:* For more simplicity let us introduce the notation:

$$\nabla = \min\left\{a\_i\right\}, \Delta = \max\left\{d\_i\right\}, \ i = 1, n. \tag{21}$$

*A* ¼ f g *A*1, *A*2, *A*3, *A*4, *A*<sup>5</sup> , (18)

min f g *ai* ; max f g *di* ½ �, *i* ¼ 1, *m:* (19)

(20)

Therefore, by constructing the linguistic variable *A*, we have satisfied condition (a). Condition (b) is also fulfilled: the polar characteristics are "the degree of risk is negligible" and "the degree of risk is extreme." To fulfill condition (c), it is necessary to build a profile, i.e., fuzzy set describing the linguistic variable *A*. We construct the membership function of profile *A* in several stages. Here we will give a description of the stages in a general form; the reader will clarify the

**Stage 1.** Let us evaluate the confidence of risk degrees of the linguistic variable *A*

*A*<sup>3</sup> � [*k*2, *k*3], *A*<sup>4</sup> � [*k*3, *k*4], and *A*<sup>5</sup> � [*k*4,100]. Here 0 < *k*<sup>1</sup> < *k*<sup>2</sup> < *k*<sup>3</sup> < *k*<sup>4</sup> < 100. **Stage 2**. Since expert estimates are given in the form of trapezoidal fuzzy numbers, first of all, it is necessary to determine the boundaries of the scale of expert estimates for each characteristic of the list from *A*. Since *m* experts take part in the assessment process, we have *m* trapezoidal fuzzy numbers. It seems reasonable to take the following boundaries of the scale: the left one is the minimum, and

on the percentage scale (0–100)% as follows: *A*<sup>1</sup> � [0, *k*1], *A*<sup>2</sup> � [*k*1, *k*2],

the right maximum of the abscissas of all the vertices of *m* trapezoids, i.e.:

**Stage 3.** Now we will establish the conformity between the intervals of the percentage scale and the trapezoidal fuzzy numbers. Geometrically, the percentage scale that corresponds to five trapezoidal fuzzy numbers, may, for example, look

*<sup>A</sup>*~<sup>1</sup> <sup>¼</sup> ð Þ 0, 0, *<sup>k</sup>*1, *<sup>t</sup>*<sup>2</sup> ; *<sup>A</sup>*~<sup>22</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*1, *<sup>k</sup>*1, *<sup>k</sup>*2, *<sup>t</sup>*<sup>3</sup> ; *<sup>A</sup>*~<sup>3</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*2, *<sup>k</sup>*2, *<sup>k</sup>*3, *<sup>t</sup>*<sup>4</sup> ; *<sup>A</sup>*<sup>~</sup> <sup>4</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*3, *<sup>k</sup>*3, *<sup>k</sup>*4, *<sup>t</sup>*<sup>5</sup> ;

system for expert estimates, the following mappings should be performed ½ �! 0, 100 min f g *ai* , max f g *di* ½ �*:* Thus, we moved the origin from point (0,0) to point ( min f g *ai* , 0) and point (100,0) to point max f g *di* ð Þ , 0 *:* For more simplicity

To transform the coordinate system of the percentage scale to the coordinate

∇ ¼ min f g *ai* , *Δ* ¼ max f g *di* , *i* ¼ 1, *n:* (21)

where:

*Banking and Finance*

like this (**Figure 4**):

*<sup>A</sup>*~<sup>5</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*4, *<sup>k</sup>*4, 100, 100 ,

**Figure 4.**

**190**

let us introduce the notation:

numbers *k, t* are appointed by experts.

*Conformity between the percentage scale and the trapezoidal fuzzy numbers.*

*A*<sup>1</sup> – the degree of risk is negligible. *A*<sup>2</sup> – the degree of risk is low. *A*<sup>3</sup> – the degree of risk is medium. *A*<sup>4</sup> – the degree of risk is high. *A*<sup>5</sup> – the degree of risk is extreme.

specifics on a practical example in the next section.

It is easy to see that the coefficient of proportionality between the abscissas of the primary and the new coordinate system is

$$
\lambda = \mathbf{0}.01(\Delta - \nabla). \tag{22}
$$

(23)

Thereby, the coordinates of the original trapezoidal fuzzy numbers will change as follows:

$$\begin{aligned} \tilde{A}\_1 &= (\nabla, \nabla, \lambda \mathbb{k}\_1 + \nabla, \lambda \mathbb{k}\_2 + \nabla); \tilde{A}\_2 = (\lambda \mathbb{t}\_1 + \nabla, \lambda \mathbb{k}\_1 + \nabla, \lambda \mathbb{k}\_2 + \nabla, \lambda \mathbb{t}\_3 + \nabla); \\ \tilde{A}\_3 &= (\lambda \mathbb{t}\_2 + \nabla, \lambda \mathbb{k}\_2 + \nabla, \lambda \mathbb{k}\_3 + \nabla, \lambda \mathbb{t}\_4 + \nabla); \tilde{A}\_4 = (\lambda \mathbb{t}\_3 + \nabla, \lambda \mathbb{k}\_3 + \nabla, \lambda \mathbb{k}\_4 + \nabla, \lambda \mathbb{t}\_5 + \nabla); \\ \tilde{A}\_5 &= (\lambda \mathbb{t}\_4 + \nabla, \lambda \mathbb{k}\_4 + \nabla, \mathbb{k}\_7, \mathbb{d}\_7). \end{aligned}$$

So, condition (c) is also satisfied.

We continue the description of the implementation of the proposed approach. Based on Algorithm 1, we find the value of the representative of the finite collection of the trapezoidal fuzzy numbers for each parameter. Consider a representative calculated for the *i-*th parameter. The risk assessment threshold value is assigned by the manager (group of managers) of the lender. It may be that different criteria thresholds will be set for different cases, for example, for one parameter, "no more than *A*2—the degree of risk is low," and for the other "no more than *A*3—the degree of risk is medium."

In general, if *Aj* ∈ *A* (see Eq. (18)) is taken as the threshold criteria value of the parameter, then credit risk is acceptable if the following addition condition is fulfilled:

$$
\tilde{R}\_i \le \tilde{A}\_j, \ i = \overline{1, n}, \ j = \overline{1, 5}. \tag{24}
$$

Here *A*~ *<sup>j</sup>* is the number corresponding to the characteristic *Aj*, while *R*~*<sup>i</sup>* is the result of aggregation of the finite collection of expert estimates of the *i-*th parameter (see Algorithm 1).

Let us summarize the above as a generalized algorithm. So, we have the following input: *m* expert estimates of *n* parameters out of the set *P* ¼ *pi* , *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, the result of aggregation *R*~*<sup>i</sup>* of the finite collection of expert estimates for this parameter, the threshold criteria value *Ak*, *k* ¼ 1, 5 selected from Eq. (18), and coordinates *k, t* specified by the manager (group of managers) of the lender for use in Eq. (23).

#### **Algorithm 2.**

Step 0: *Initialization: fix pi* <sup>∈</sup> *<sup>P</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, *the result of aggregation <sup>R</sup>*~*<sup>i</sup> of the finite collection of expert estimates R*~ *<sup>j</sup>* , *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … *for this parameter, the threshold criteria value At*, *t* ¼ 1, 5, *coordinates*: *k*1*, … ,k*4; *t*1*, … ,t*5.

Step 1: *Compute* ∇, *Δ by Eq.* (21) *and λ by Eq.* (22).

Step 2: *Compute <sup>A</sup>*~*t*, *<sup>t</sup>* <sup>¼</sup> 1, 5 *by Eq*. (23).


**Summary.** The section looks at the realization of the proposed approach. The linguistic variable "degree of credit risk," polar percentage and coordinate scales are formed. The criterion for an assessment of the credit risks is generated. This section also presents two generalized algorithms for implementing the proposed approach.
